Interface frictional anisotropy of dilative sand

Understanding direction-dependent friction anisotropy is necessary to optimize interface shear resistance across soil-structure. Previous studies estimated interface frictional anisotropy quantitatively using contractive sands. However, no studies have explored how sand with a high dilative tendency around the structural surface affects the interface shear response. In this study, a series of interface direct shear tests are conducted with selected French standard sand and snakeskin-inspired surfaces under three vertical stresses (50, 100, and 200 kPa) and two shearing directions (cranial → caudal or caudal → cranial). First, the sand-sand test observes a higher dilative response, and a significant difference between the peak and residual friction angles (ϕpeak − ϕres = 8°) is obtained at even a lower initial relative density Dr = 40%. In addition, the interface test results show that (1) shearing against the scales (cranial shearing) mobilizes a larger shear resistance and produces a dilative response than shearing along the scales (caudal shearing), (2) a higher scale height or shorter scale length exhibits a higher dilative tendency and produces a higher interface friction angle, and (3) the interface anisotropy response is more pronounced during cranial shearing in all cases. Further analysis reveals that the interface friction angle and dilation angle are decreased with the scale geometry ratio (L/H). For L/H values between 16.67 and 60, the interface dilation angle varies between 9° and 4° for cranial first shearing and 3.9°–2.6° for caudal first shearing. However, the difference in dilation angle within the same shearing direction is less than 1°.

Table 1.Geometrical parameters of bio-inspired surfaces: (a) characteristics of surfaces tested in this study; (b) illustration of a snakeskin-inspired surface and two-way shearing.N indicates the number of surface scales.The total length of surface scales is fixed as 72 mm in all plates.Loading Plate The vertical stress applied through the loading rod mounted on a steel ball and loading plate in the conventional apparatus is replaced with a loading plate attached to a vertical loading frame.Note that the loading plate and rod were not fixed in the conventional apparatus.
Shearing System The linear motion guide (LM guide) is installed for smooth and frictionless movement of the lower shear box bolted to the outer moving box.The lower shear box is the platform over which the snakeskininspired surface is fastened to perform the interface tests.(c) Grain size distribution curve (GSD) of selected particle size range II.Based on the SEM image II, the shape of selected particles can be classified as angular, and the grain size distribution curve shows a uniform particle size range with C u close to 1.44 (refer to Table 2).Table 2. Basic properties of selected French standard sand (FSS) used in this study.a The maximum and minimum void ratios are measured according to 24,25 .b Direct shear test using a 2.5-inch circular shear box.The dilation angle is determined using Taylor ' s flow rule 26 at single vertical stress of 100 kPa.Loading System The vertical loading system uses an air cylinder.Additionally, the pressure gauge is installed to monitor the applied air pressure.The shear motor is used to easily control the constant shear strain rate.
Sensor System The vertical and horizontal loads applied to the specimen are measured using each load cell.In particular, a reaction arm is installed to connect the fixed block with the upper shear box.It facilitates transferring the friction force generated from the soil-plate interface to a horizontal load cell.Two LVDT sensors are installed to measure vertical and horizontal displacements.

Experimental methods
A total of 39 interface direct shear tests are performed on six snakeskin-inspired surfaces and one untextured surface under three initial vertical stresses (50, 100, and 200 kPa) and two-way shearing directions.The two-way shearing consists of (1) cranial → caudal test: cranial shearing direction during the first half cycle and then caudal shearing during the second half cycle and (2) caudal → cranial: caudal direction during the first half cycle and cranial shearing during the second half cycle (refer Table 1b).
Sand specimen is air-pluviated over the snakeskin-inspired surface in the shear box for targeting an initial relative density of 40%.Based on the 10% shear strain failure criterion, the sand specimen is sheared until a displacement of 6 mm in a circular shear box for the sand-sand test and of 10 mm for the sand-snakeskin inspired interface tests under a constant 1 mm/min shear rate.All sensors are connected to a data logger to record and save data automatically.Also, the LabView program is used for the initial setting of sensors and continuous monitoring.The shear stress is calculated by dividing the measured shear force by the cross-sectional area of the specimen.All test results including the interface friction (ϕ peak ) and dilation angle (ψ) are summarized in Table .3.

Shear behavior of French standard sand
Figure 3 presents the soil-soil shear response under three initial vertical stresses (50, 100, and 200 kPa).The shear responses are plotted against the horizontal displacement related to the shear strain.During the initial shearing process, the change in the vertical stress follows the trend of vertical displacement associated with the volume change, yet it nearly remains constant after dilation (Fig. 3a).The shear stress-horizontal displacement response shows an increase in the initial stiffness (not shown here but can be indirectly assessed from the stress-displacement response) and peak shear resistance with the increase in vertical stress.The strain softening behavior is more pronounced at higher vertical stress.In addition, an increase in the vertical displacement occurs with an increase in the vertical stress.The volumetric dilatancy occurs after the initial contraction at 50 and 100 kPa (Fig. 3c).In summary, the dilative response is observed and a significant difference between the peak and residual friction angles (ϕ peak -ϕ res = 8°) is obtained at even a lower initial relative density D r = 40%.These observations confirm the fact that the FSS particles can be classified as dilative sand.
Figure 2. Schematic of a modified direct shear apparatus modified from 22,34 .For the interface tests and to accommodate snakeskin-inspired surfaces, the vertical stress application through the loading rod mounted on a steel ball and rectangular loading plate in the conventional apparatus is replaced by a rectangular loading plate attached to a vertical loading frame.For soil-soil tests, a circular shear box with a circular loading plate fixed to the vertical loading rod is used.Note that the loading plate and loading rod were not fixed in the conventional apparatus.

Interface shear behavior of snakeskin-inspired surfaces
Figure 4 shows the interface frictional anisotropy under three vertical stresses with the same scale geometry (L = 12 mm, H = 0.3 mm).Slight fluctuation of vertical stress takes place during cranial shearing process: the cranial shearing produces slight increase in the vertical stress while the caudal shearing keeps the vertical stress constant.This is because the interface shearing resistance during the cranial shearing process propagates toward the loading plate and eventually increases the local mean effective stress (Fig. 4a,d).Figures 4b and e present shear stress-shear displacement response.Higher vertical stress produces larger shear stress, which is nearly similar for both the cranial → caudal and the caudal → cranial tests in the same shearing direction.While the cranial first shearing direction mobilizes higher shear resistance, the subsequent caudal shearing reduces shear stress.Conversely, caudal first shearing mobilizes lower shear resistance, and cranial second shearing enhances shear stress.Regardless of the starting direction, cranial shearing results in higher shear stress in all cases.A previous study 5 explored the effects of shearing direction and scale geometry on induced soil deformation by analyzing particle image velocimetry.Shearing in the cranial direction enables the soil to latch onto the scales and increases the contact area of particles behind the scales.Eventually, the wedge-shaped shear failure mode tends to develop at the leading front of the scales, and shear localization propagates around the scales.However, the caudal shearing direction hinders the latching of the soil on the scales and fails to enhance the contact area.Accordingly, the induced failure deformation produces shear bands that are evenly propagated along the scale apex.As shown in Fig. 4c,f, the cranial first shearing process produces a more pronounced dilative tendency and volume change is larger for the cranial → caudal test than for the caudal → cranial test.This is because the shearing process in the cranial direction causes dilation of the soil at the leading end of the scales and contraction at the trailing end.

Effect of scale geometry
Figure 5 shows the results of the interface shear behavior by varying the scale height while keeping the scale length constant at 12 mm.A higher scale height mobilizes a larger shear stress in the cranial shearing direction, yet the difference of shear stress in the caudal shearing direction is less pronounced with scale height (Fig. 5a,c).The scale height changes the tendency of the vertical displacement associated with the volumetric response, as shown in Fig. 5b,d.For the cranial first shearing direction, a surface with a lower scale height (H = 0.1 mm) exhibits contractive behavior, whereas the responses at relatively higher scale heights (H > 0.1 mm) are dilative.The less dilatancy occurs for the caudal first shearing.This is because a higher height increases the interface roughness and develops larger individual passive wedges with the dilation of the soil at the front of the scale and the contraction of the soil behind the scale.Figure 6 shows the response of the interface shear behavior by varying the scale length at the same scale height (H = 0.3 mm).Note that shorter scale length indicates more number of surface scales (but the total length remains the same).A shorter scale length mobilizes higher shear resistance during the cranial shearing direction, yet the caudal direction shows less change of shear stress with scale length (Fig. 6a,c).The dilative tendency is observed for all the scale lengths for the cranial → caudal test.But, contractive behavior is more pronounced for the caudal → cranial test except for the shorter scale length (L = 6 mm) as shown in Fig. 6b,d.This is because more textured surfaces (i.e., shorter scale lengths) mobilize more individual wedges, leading to a more dilative overall volumetric response during the cranial shearing process.

Evolution of interface friction angle
The interface friction angle is quantified to explore the effect of scale geometry and the shearing sequence.Figure 7a shows the evolution of the interface friction angle for snakeskin-inspired surfaces with varying scale heights at a constant scale length (L = 12 mm) under two shearing directions.Compared to the untextured surface, the textured surfaces produce higher interface frictional resistance in all cases.The interface friction angle is increased with scale height, and cranial shearing produces a higher interface friction angle than caudal shearing.
For the cranial first shearing case, the difference between the friction angle at 0.1 mm and 0.72 mm scale heights is 19.7°, and for the cranial second case is 20°.A sharp increase in interface friction angle takes place within scale height between 0.1 and 0.3 mm during the cranial starting direction.Subsequently, an increase in height results in a moderate increase in the interface friction angle.A previous study explored the effect of surface roughness on sand-steel interface behavior 35 .The increase in the surface roughness related to the scale height produced a higher interface frictional strength, yet it reached an asymptotic value at a certain scale height.The trend of the interface frictional resistance is similar to the cranial shearing direction observed in this study.Meanwhile, the caudal shearing direction exhibits a slight increase in the interface friction angle, regardless of its sequence.The difference between the interface friction angles at 0.1 and 0.72 mm scale height is 4.7° and 5.6° for the caudal first and second cases, respectively.Figure 7b shows the evolution of the interface friction angle for the snakeskin-inspired surfaces with varying scale lengths at a constant scale height (H = 0.3 mm) under two shearing directions.During cranial shearing, the interface friction angle dramatically decreases with increasing scale length.The differences between the friction angles at L = 6 mm and L = 24 mm scale lengths are 11.7° and 13.7° for the cranial first and second cases, respectively.Similarly, the difference in the interface angles is 3° for the first caudal case and 2.9° for the second caudal case.Cranial shearing direction produces higher interface friction angle regardless of the shearing sequence.
Indeed, the changing trend of interface friction angle is affected by the starting shear direction while the shearing sequence has no effect on the interface friction angle.Previous studies 17,22 revealed that volumetric contraction at the trailing end of the scale during the starting shearing densified soil around the scale, difference in the interface friction angle between the cranial and caudal shearing directions was pronounced depending on the starting shearing direction, and thus the caudal first-caudal second shearing sequence produces larger friction difference than the cranial first-cranial second shearing sequence.However, the current study observes that the interface friction angle in the first shearing cycle is almost identical to that in the second shearing cycle, and thus densification effect during the first shearing is diminished.This is because angular particles during the first shearing cycle produce less volumetric contraction behind the scale, and the subsequent second shearing cycle does not cause a significant change in the mobilized shear resistance.

Discussion and implications to geotechnical application
The normalized scale geometry ratio, which is defined as the ratio of the scale length to height, is adopted to further analyze the interface frictional anisotropy.Figure 8 shows the interface friction angle as a function of the L/H ratio.Note that a large L/H indicates a lower scale height at the same scale length.Compared to caudal shearing, cranial shearing produces a higher interface friction angle at all scale geometry ratios regardless of the shearing sequence.In particular, the interface friction angle in the same shearing direction for the first shearing cycle (i.e., first cranial or first caudal) is almost identical to that for the second cycle of shearing (i.e., second cranial or second caudal) because of the unique dilative characteristics of the FSS particles, as discussed in the previous section.Interestingly, it is also observed that the shearing direction has a greater effect on the interface friction angle in regions with a lower L/H ratio.
Further analysis is conducted to investigate the evolution of the interface dilation angle across sandbioinspired plate.It is selected for experimental cases of strain-softening response that show a significant distinction between the peak and residual shearing stresses for both the cranial → caudal and caudal → cranial tests.Then, the Taylor's flow rule is applied as follows 26 .
(1) where ψ is the interface dilation angle [°], τ peak is the peak shear stress [kPa], τ res is the residual shear stress [kPa], and σ′ v is the vertical stress [kPa].Figure 9 presents the interface dilation angles against different shearing directions and scale geometry ratios.The interface dilation angle decreases with the increase in the scale geometry ratio.For L/H values between 16.67 and 60, the interface dilation angle varies between 9°-4° for cranial first shearing and 3.9°-2.6°for caudal first shearing.Similar to the interface friction angle, the dilation angles for caudal shearing cycles are nearly identical regardless of the shearing sequence (i.e., first and second caudal).However, there is small difference (less than 1°) in the dilation angles for the cranial shearing cycles (first and second cranial).Indeed, this analysis confirms that the FSS dilates during the first shearing cycle and does not show further increase or changes in dilation during the second shearing cycle, which eventually leads to no considerable change in the mobilization of the dilation angle during the second shearing cycle.The stress-dilatancy equation was proposed by Bolton 36 .The empirical  Indeed, the difference between peak fiction angle and residual friction angle is related to a dilatancy coefficient.The empirical equation is re-evaluated by comparing with the results obtained from this study.Figure 10 shows a strong correlation between the predicted and observed peak interface friction angles.The observed peak interface friction angle exhibits high accuracy, with a substantial R 2 value of 0.994, along with a low standard deviation (S.D).
The interface friction angle and dilation angle of dilative sand obtained from various scale geometries and different shearing directions help to understand the interface shear and dilation response, and ultimately optimal interface friction angle can be selected for the design of bioinspired geo-structures (e.g., driven piles, offshore onopoles, soil anchors, and tunnel boring machines) to achieve desired frictional anisotropy.For example, if the surface of a driven pile is textured with snakeskin-inspired scales having relatively higher L/H scale ratio, the minimized mobilized shear resistance could be achieved during the installation.The outcome of this study is not limited to snakeskin-inspired surfaces but is also applicable to other geotechnical infrastructure related interface shearing systems such as textured geomembranes and ribbed soil reinforcements.In this study, a lower initial relative density (D r = 40%) is selected.However, note that change in relative density influences shear behavior.At the given scale geometry, higher initial relative density produces higher interface friction resistance, provided there is no particle crushing or damage to the interface material.Future studies are required to conduct in-situ field testing of textured bioinspired surfaces, incorporating data monitoring, and assessing real-world conditions to ensure practical effectiveness in diverse applications.

Figure 1 .
Figure 1.The particle size characterization of French standard sand (FSS): (a) SEM images of three particle size ranges of FSS; (b) Grain size distribution curve (GSD) of FSS; (c) Grain size distribution curve (GSD) of selected particle size range II.Based on the SEM image II, the shape of selected particles can be classified as angular, and the grain size distribution curve shows a uniform particle size range with C u close to 1.44 (refer to Table2). https://doi.org/10.1038/s41598-024-56621-1www.nature.com/scientificreports/

Figure 3 .
Figure 3. Soil-Soil shear response obtained from a modified circular direct shear apparatus under three different vertical stresses: Horizontal displacement against (a) vertical stress; (b) shear stress; (c) vertical displacement.

Figure 4 .
Figure 4. Response of interface frictional anisotropy using modified rectangular direct shear apparatus under three different vertical stresses: Horizontal displacement against (a,d) vertical stress; (b,e) shear stress; (c,f) vertical displacement.Experimental case is L = 12 mm and H = 0.3 mm.

Figure 5 .
Figure 5.The effect of scale height on the interface shear response: Horizontal displacement against (a,c) shear stress; (b,d) vertical displacement.The applied vertical stress is 100 kPa and scale length L is fixed as 12 mm.

Figure 6 .
Figure 6.The effect of scale length on the interface shear response: Horizontal displacement against (a,c) Shear stress; (b,d) vertical displacement.The applied vertical stress is 100 kPa and scale height H is fixed as 0.3 mm.Note that a shorter scale length indicates a greater number of surface scales.

Figure 7 .
Figure 7. Interface friction angle as a function of snakeskin-inspired scale geometry under two shearing directions: (a) scale height (L = 12 mm fixed); (b) scale length (H = 0.3 mm fixed).

Figure 8 .
Figure8.The interface friction angle as a function of normalized scale geometry L/H ratio.The interface friction angle significantly varies between cranial and caudal shearing directions at the given scale geometry ratio.However, the interface friction angle for similar shearing directions in a two-way shearing process (first cranial and second cranial, first caudal and second caudal) is almost identical.

Figure 9 .
Figure 9.The dilation angle as a function of normalized scale geometry L/H ratio.The dilation angle is determined using the Taylor's flow rule at 100 kPa vertical stress.

Figure 10 .
Figure 10.Comparison between the observed (this study) and the predicted interface friction angle.The empirical equation (ϕ peak = ϕ res + 0.8ψ) proposed by Bolton 36 is used to predict the interface friction angle.Solid line shows the central trend defined by a 1:1 line.Dotted lines show ± 1 standard deviation from the central line.

Table 3 .
26mmary of interface friction strength obtained from modified direct shear apparatus with various scale geometries, three vertical stresses, and two-way shearing: (a) Cranial → Caudal; (b) Caudal → Cranial.The interface friction angles for varying scale geometries are quantified by selecting the peak shear stress at the corresponding vertical stress and shearing direction based on the Mohr-Coulomb failure criterion.The correlation coefficient R 2 is above 0.98 in all the cases.The dilation angles are measured at 100 kPa vertical stress using Taylor's flow rule26.